Fault Estimations and Non-Fragile Control Design for Fractional-Order Multi-Weighted Complex Dynamical Networks

This paper deals with the robust fault estimation based synchronization problem for a class of fractional-order multi-weighted complex dynamic networks subject to external disturbances. Specifically, the synchronization problem is converted to an equivalent uniformly ultimately boundedness problem of the corresponding error system by use of spectral decomposition of the Laplacian and proper scalings of the faults. First, by introducing an intermediate estimator technique and using monochromatic Lyapunov function method, the sufficient condition for uniformly ultimately boundedness of the resulting error system via fault estimation based non-fragile controller is presented. Next, by using the linear matrix inequality (LMI) technique and optimization approach, the existence condition in the form of LMIs for designing the fault estimation based robust non-fragile controller is derived. Finally, two numerical examples including financial model are provided to demonstrate the validity and feasibility of the developed theoretical results.


I. INTRODUCTION
Recently, complex dynamical networks (CDNs) have been applied in various areas such as transportation networks, power grids, unmanned aerial vehicles, image processing and so on [1]- [8]. Precisely, most of the existing results are formulated as CDNs with single weight, but for actual application, real-time networks might be represented by CDNs with multi-weights, such as the transportation networks, electricity distribution networks, aviation networks and social networks. Currently, few tremendous results for the synchronization criterion of CDNs with multiple weights have been explored, for instance, [9]- [11]. In [10], the authors derived a criterion for the passivity analysis of multi-weighted CDNs (MWCDNs) on the basis of edges-based pinning adaptive control strategy and Lyapunov stability theory. More recently, the fractional-order dynamical systems have been widely utilized in various fields of science and engineering due to its lower order, less parameters and higher accuracy [12]- [16].
The associate editor coordinating the review of this manuscript and approving it for publication was Jun Hu . In [14], the authors investigated the global synchronization problem for fractional-order CDNs with discontinuous nodes via finite-time approach. However, because of increasing difficulty of the fractional-order CDNs, it is not easy to ensure the synchronization criteria. So, it is necessary and significant to develop a more reliable control algorithm for the synchronization of fractional-order multi-weighted complex dynamic networks. The problem of robust synchronization of fractional-order CDNs with unknown bounded uncertainty and time delay, has been discussed in [17], where the fractional uncertainty and disturbance estimator approach has been presented to obtain the required theoretical result.
As we know, in the procedure of controller execution in many real-time systems, it is hard or even not possible to attain accurate controller due to the existence of some inevitable uncertainties in control design parameters. Further, it is proved that the vanishingly small perturbations in control parameters can produce poor performance or damage the control system. Therefore, it is significant to design a controller that can admit uncertainties and disturbances in its VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see http://creativecommons.org/licenses/by/4.0/ design. This inspires the study of non-fragile control problems for CDNs in the recent decades [18]- [20]. On the other hand, the fault estimation method plays an important role in the development of synchronization criteria for CDNs, whose purpose is to track the fault signals occurred in the network and give their exact sizes by constructing an appropriate fault estimator. Recently, very few results about the fault estimation problem for CDNs can be found in [21]. In [21], the authors investigated an integrated fault estimation and accommodation problem for a class of CDNs via distributed adaptive estimator approach. In [22], the joint state and fault estimation problem of a nonlinear stochastic systems in the presence of sensor saturations and randomly occurring faults has been reported. However, to the best level of authors' knowledge, distributed intermediate fault estimation based synchronization problem for fractional-order MWCDNs via non-fragile control law has not been reported yet in the existing works. Motivated by the aforementioned works, in this paper we focus on the design of a fault estimation based non-fragile control for synchronization of fractional-order MWCDNs subject to external disturbances and the lack of full-state measurements. The main features and contributions of this paper can be summarized as follows: • Based on fault estimation and distributed intermediate estimator approach, a new observer-based synchronization control scheme is proposed for fractional-order multi-weighted CDNs with external disturbances.
• Specifically, by using spectral decomposition of the Laplacian and proper scalings of the faults, the synchronization problem is converted to an equivalent simultaneous uniformly ultimately boundedness problem of the corresponding error systems.
• Simulation result reveals that the performance of the proposed non-fragile control design is effective since the distributed intermediate estimators needs only less information for communications.

II. SYSTEM DESCRIPTION AND PRELIMINARIES
In this section, first we define the Riemann-Liouville derivative, then, discuss the construction of fault estimation based non-fragile control design by employing fractional-order Luenberger-type state observer. Definition 1 [23]: The definition of fractional integral is described by where α denotes the fractional-order and (·) denotes the Gamma function defined by (α) = ∞ 0 e −z z α−1 dz. Definition 2 [23]: The Riemann-Liouville derivative is defined by Consider a class of fractional-order MWCDNs consisting of N identical nodes with actuator faults and external disturbances, where the dynamics of the i-th node is represented by the following nonlinear descriptor equations: where D α t denotes the Riemann-Liouville derivative; 0 < α < 1 is the fractional order; x i (t) ∈ R n is the state vector of the ith node; y i (t) ∈ R q is measured output vector of the i-th node; u i (t) ∈ R m denotes the control input vector of the i-th node; f i (t) ∈ R s is the fault signal of the i-th node; ω i (t) is the network external disturbance which belongs to L 2 [0, ∞); φ i (t 0 ) is the continuous initial vector function; g(t, x i (t)) is a vector-valued time-varying nonlinear function, which satisfies bounded sector constraint which will be defined later; A 1 , A 2 , B, C, D and E are known constant matrices with appropriate dimensions; a v ∈ R + (v = 1, 2, . . . , r) denote the coupling strength of the vth coupling form; v ∈ R n×n represent the positive diagonal inner coupling matrix of the vth coupling form; Let e i (t) = x i (t) − s(t) be the synchronization error, where s(t) ∈ R n represent the state trajectory of the unforced target node and which satisfies t 0 D α t s(t) = A 1 s(t) + Bg(t, s(t)). Further, y s (t) = A 2 s(t) is the unforced isolate output vector. Then, using (1), the dynamic equations of the fractional-order error system can be expressed as follows: Further, let us define the intermediate variable as follows: where η i (t) = C * Df i (t) in which C * is any matrix with appropriate dimension such that (I − CC * )D = 0 and δ is a known positive scalar. Then by using (2) and (3), it can be written as In this study, the state information of all nodes are assumed to be unknown. To achieve synchronization, state and fault signal estimation, the intermediate state estimators of the systems (2), (3) and (4) are taken respectively in the following form [24]: and η i (t), respectively;ŷ i (t) ∈ R p denotes the output estimation vector of the i-th node; L ∈ R n×p is the gain matrix of the observer. Let us introduce some error vectors Then the overall dynamics of the distributed intermediate estimators can be represented by where G(t, ξ i (t)) =ḡ(t, e i (t)) −ḡ(t,ê i (t)). In this paper, we are interested in designing of the intermediate estimator and non-fragile state-feedback controller are expressed as follows: where K is the feedback controller gain matrix which to be determined in the forthcoming section, K (t) denotes the control gain matrix with the structure where M and N are known real constant matrices and (t) is an unknown time-varying matrix satisfying T (t) (t) ≤ I . Now, by substituting the control law (13) into (2), we can obtain as follows: By the virtue of Kronecker product and the systems (9), (10) and (14) are expressed as follows: where This study aims at designing a non-fragile controller, intermediate estimator in the form of (13) to fractional-order MWCDNs (1) with actuator faults. To achieve this goal, it is sufficient to show that the closed-loop systems (15)- (17) is asymptotically synchronized. Before presenting the main results, we provide some preliminaries, which are more indispensable for the later development.
Assumption 1: For the nonlinear function g(t, x(t)), there exists a known real constant matrix V such that: for any x(t) ∈ R n , where · refers to the Euclidean vector norm.
Lemma 1: [25] The fractional-order nonlinear differential equation D α t x(t) = f (x(t)) can be written as

III. MAIN RESULTS
In this section, we will design a non-fragile controller (13) to achieve synchronization for fractional-order MWCDNs (1). First we show that if the control gain is known then the closedloop system (15)- (17) is uniformly bounded. Theorem 1: Let Assumptions 1 and 2 be true. For given positive scalars δ, , ν and positive diagonal matrices V 1 and V 2 , the closed-loop system (15)-(17) is uniformly bounded if there exist symmetric positive matrices P 1 , P 2 and P 3 , and a positive scalar such that the following matrix inequality holds:¯ 3 Proof: It follows from Lemma 1 in [25] that the fractional-order MWCDNs (15)- (17) can be written as follows: T . Then, we choose the Lyapunov function for the systems (19) (22) This Lyapunov function integrates all the monochromatic υ a (w, t)(a = 1, 2, 3) with a weighting function ζ (w) on the whole spectral range. Now, by calculating the time derivative of (22) along the solution of the systems (19)-(21), we can geṫ From Assumption 2, it can be obtained that there exist scalar κ such that D α t η(t) ≤ κ. By considering this fact, for any chosen positive scalar ν. Then, the following inequality always hold: Moreover, according to Assumption 1, we can obtain the following inequalities where V 1 and V 2 are known positive matrices. By considering (23)- (26), it is easy to obtain thaṫ where ζ (t) = e T (t) ξ T (t) T (t)ḡ T (t, e(t)) G T (t, ξ T (t)) ω T (t) T ,˜ = + ϑ (t)υ + (ϑ (t)υ) T and the elements of , ϑ and υ are defined in the theorem statement. Moreover, based on Lemma 2 in [26], for any scalar > 0, the RHS of (27) can equivalently be written as Based on the Schur complement, it is noted that RHS of (28) is equivalent to the LHS of LMI of (18). Hence, it follows from (18) that˜ < 0. On the other hand, it is noted that V (t) ≤ max{λ max (P 1 ), λ max (P 2 ), λ max (P 3 )}( ξ (t) 2 + (t) 2 + e(t) 2 ). Now let us denote˜ = −¯ , it is obvious that if¯ < 0 i.e.,˜ > 0, theṅ where = λ min (˜ ) χ , χ = max{λ max (P 1 ) ξ (t) 2 + λ max (P 2 ) (t) 2 + λ max (P 3 ) e(t) 2 } and = νκ 2 . Moreover, we consider the set W = ξ (t), (t), e(t) : If ξ (t), (t), e(t) ∈W, whereW is the supplementary set of W, then it follows that From the inequalities (29) and (30), it can be observed thaṫ Obviously, the pair ξ (t), (t), e(t) is uniformly bounded and exponentially converges to the set W at a rate greater than e − t from (29). Hence, the proof is completed.
It should be noted that if the controller gain matrix are unknown, then the constraint in (18) cannot be solved directly via MATLAB LMI control toolbox due to the existence of nonlinear terms. To rectify this problem, we apply linear congruence transformation to the derived conditions in the Theorem 1. The following theorem presents the necessary LMI constraints to attain the control gain matrix.
Theorem 2: Suppose that Assumptions 1 and 2 hold. Then the intermediate estimator approach based non-fragile statefeedback controller can ensure that all the signals in the closed-loop systems (15)- (17) are uniformly bounded for given positive scalars δ, , ν, λ and positive diagonal matrices V 1 and V 2 , if there exist symmetric positive matrices P 1 , P 2 , P 3 , any suitable dimensioned matrices X , Y and L, and a positive scalar such that the following LMIs are satisfied:  9,9 = −ν(I ⊗ I ) and the remaining elements ofˆ are same as of is defined in the statement of Theorem 1 and all other elements are zero. Moreover, the non-fragile state feedback control design parameters can be obtained by the following relations: K = YX −1 and L = P −1 1 L. Proof: In order to provide the proof for this theorem, we consider the linear congruence transformations P 3 C = CX , Y = XK and L = P 1 L. Then, by applying Schur complement, the inequalities in (18) can be equivalently written as (31). It is noted that the assumption P 3 C = CX is not a linear inequality and so it is difficult to solve via MATLAB LMI toolbox. To resolve this difficulty, P 3 C = CX can be replaced by relatively equivalent inequality constraint VOLUME 8, 2020 (32) for any small positive scalar λ. Hence, the proof is completed.
Remark 1: It is worth pointing out that, a great number of research works regarding the issue of synchronisation of CDNs have been reported in the recent literature, for instance see [3]- [8]. Nevertheless, only very few works have been focused on the problem of synchronisation of fractionalorder CDNs with external disturbances [16], [17]. It should be noted that, all the aforementioned works do not consider multi-weights in the system. Very recently, some interesting results on synchronisation of fractional-order CDNs with multi-weights have been discussed in [12]. However, the issue of the robust fault estimation based non-fragile control design for fractional-order CDNs subject to multi-weights, actuator faults and external disturbances has not yet been discussed in the literature. Thus, the main contribution of this paper is to fill such a gap through employing a non-fragile control law based on intermediate estimator for achieving robust synchronization in fractional-order CDNs in the presence of multi-weights and external disturbances, which makes this work different form the existing works on fractional-order CDNs.

IV. NUMERICAL EXAMPLES
In this section, two numerical examples including the financial model are given to demonstrate the effectiveness and superiority of the proposed non-fragile controller design (13).
Example 1: Consider the fractional-order MWCDNs (1) with four nodes and three coupling weights and its parameters are taken as follows: The inner coupling of non-delay matrices are selected as follows: Further, the nonlinear function is chosen as Then, by solving the LMIs (31) and (32) in Theorem 2 via MATLAB LMI toolbox, we can obtained the set of feasible solution from which the non-fragile state feedback control and observer gain matrices are given by 39518 VOLUME 8, 2020 For the simulation purposes, the fault f i (t) (i = 1, 2, 3, 4) are chosen as follows:  and the external disturbance of all four nodes, the initial conditions for the states of nodes and the isolated node are respectively chosen as follows: ω i (t) = [0.25 sin(π t) 0.30 sin(5π t) VOLUME 8, 2020 FIGURE 5. Fault η 2 (t ) and its estimateη 2 (t ). The fractional-order financial chaotic system is borrowed from [27], which is described as follows: where T is the state variable in which x i1 (t) represents the interest rate; x i2 (t) denotes the investment demand and x i3 (t) is the price index; and positive constants δ 1 , δ 2 and δ 3 represent the saving amount, per-investment cost and elasticity of demands for commercials, respectively. Here, the system parameters are taken as follows: δ 1 = 3, δ 2 = 0.1, and δ 3 = 1. Now, we show that our proposed control can be applied to fractional-order financial system (33), which contains six identical nodes (N = 6) and four coupling weights (v = 1, 2, 3, 4) given by where the fractional-order Let the fault f i (t) (i = 1, 2, 3, 4, 5, 6) are selected as given below; 4)), elsewhere, Then, by solving the LMIs (31) and (32) in Theorem 2 via MATLAB LMI toolbox, we can obtain a set of feasible solutions from which the non-fragile state feedback control and observer gain matrices are computed by       and 8(c), respectively. It can be easily observed from these figures that the states of the nodes are exactly synchronized with the states of the isolate node within a short period which demonstrates the effectiveness of the proposed controller. In Figs 9-14, the actual faults and their corresponding estimations are plotted, where it can be observed that the given signals are perfectly estimated from the beginning itself. Thus, from this simulation results, we can conclude that the proposed controller effectively synchronize the financial model even in the presence of external disturbances. Therefore, the results reveal that the proposed control design is more suitable for synchronizing the addressed financial model.

V. CONCLUSION
In this work, the non-fragile control problem has been analyzed for a class of fractional-order MWCDNs via intermediate estimator approach. Particularly, the Luenberger fractional-order state observer has been created to estimate the states of the addressed system. By using proper scalings of the faults, the synchronization problem is converted to an equivalent simultaneous uniformly ultimately boundedness problem for the corresponding error systems. Further, the uniformly ultimately boundedness criterion has been developed for the error system under consideration in terms of LMIs. At last, two numerical examples have been provided to demonstrate the effectiveness of the proposed controller. It is worth noting that the stochastic synchronization of fractionalorder CDNs with multi-weights based on the intermediate estimator approach will be the topic of our future research work.